DES S-box 4 is not like the others
DES S-box 4 is not like the others
Stian Fauskanger
University of Bergen
Department of Informatics
The Selmer Center
October 14, 2014
DES S-box 4 is not like the others
DES
Data Encryption Standard
Image source: Wikipedia.org
DES S-box 4 is not like the others
DES
CL = PH ⊕ F1 ⊕ · · · ⊕ F15
CH = PL ⊕ F2 ⊕ · · · ⊕ F16
Image source: Wikipedia.org
DES S-box 4 is not like the others
DES
CL = PH ⊕ F1 ⊕ · · · ⊕ F15
CH = PL ⊕ F2 ⊕ · · · ⊕ F16
Image source: Wikipedia.org
DES S-box 4 is not like the others
DES
CL ⊕ PH = F1 ⊕ · · · ⊕ F15
CH ⊕ PL = F2 ⊕ · · · ⊕ F16
Image source: Wikipedia.org
DES S-box 4 is not like the others
F repeats input-bits to adjacent S-boxes
16 15
Input bit:
14 13
12 11
10
9
8
7
6
5
4
3
E(X)
Key bit: 25 24
S4
23 22 21 20 19 18
17 16 15 14 13 12
S5
S6
11 10
9
8
S7
7
6
5
S8
4
L
DES S-box 4 is not like the others
F repeats input-bits to adjacent S-boxes
16 15
Input bit:
14 13
12 11
10
9
8
7
6
5
4
3
E(X)
Key bit: 25 24
S4
23 22 21 20 19 18
S5
x1
x1 ⊕ y1 = k
17 16 15 14 13 12
y1
S6
and
x2
11 10
y2
9
8
7
S7
x2 ⊕ y2 = k 0
6
5
S8
4
L
DES S-box 4 is not like the others
Distribution on XOR of 8 outputs

k1 k10 ...k8 k80  · · ·
rst
..
.
x
x = Pr(rst | k1 k10 ...k8 k80 )


DES S-box 4 is not like the others
Number of different distributions
n
1
2
3
4
5
6
7
8
Upper bound
16
40
80
140
224
336
480
660
123
16
40
80
140
224
336
480
660
234
16
40
80
140
224
336
480
660
345
16
40
80
140
224
336
480
660
456
16
24
32
40
48
56
64
72
567
16
40
80
140
224
336
480
660
678
16
40
80
140
224
336
480
660
781
16
40
80
140
224
336
480
660
812
16
40
80
140
224
336
480
660
DES S-box 4 is not like the others
Rank of distributions
n
1
2
3
4
5
6
7
8
Upper bound
6
9
13
18
24
31
39
48
123
6
9
13
18
24
30
36
42
234
6
9
13
18
24
31
39
48
345
6
9
13
18
24
29
34
39
456
6
7
8
9
10
11
12
13
567
6
9
13
18
24
31
39
48
678
6
9
13
18
24
31
39
48
781
6
9
13
18
24
31
39
48
812
6
9
13
18
24
31
39
48
DES S-box 4 is not like the others
Right and left distribution
An S-box is a mapping S(x5 , x4 , x3 , x2 , x1 , x0 ) = (y3 , y2 , y1 , y0 )
Definition
The right distribution is the distribution of (x1 , x0 , y3 , y2 , y1 , y0 ).
DES S-box 4 is not like the others
What’s special about S-box 4?
Definition
The right distribution is the distribution of (x1 , x0 , y3 , y2 , y1 , y0 ).
0
1
2
3
0
1
1
1
1
1
0
1
2
1
P
a
2
2
0
0
2
3
2
2
0
0
4
1
1
1
1
5
2
1
0
1
fr1 (c)fr2 (c ⊕ a)
6
1
1
1
1
7
0
1
2
1
8
0
1
2
1
9
1
1
1
1
10
0
2
2
0
11
1
0
1
2
12
0
1
2
1
13
2
1
0
1
14
2
1
0
1
15
1
1
1
1
DES S-box 4 is not like the others
What’s special about S-box 4?
Definition
The right distribution is the distribution of (x1 , x0 , y3 , y2 , y1 , y0 ).
0
1
2
3
0
1
1
1
1
1
0
1
2
1
P
a
2
2
0
0
2
3
2
2
0
0
4
1
1
1
1
5
2
1
0
1
6
1
1
1
1
fr1 (c)fr2 (c ⊕ a) =
7
0
1
2
1
P
a
8
0
1
2
1
9
1
1
1
1
10
0
2
2
0
11
1
0
1
2
12
0
1
2
1
f(r1 ⊕s) (c)f(r2 ⊕s) (c ⊕ a)
13
2
1
0
1
14
2
1
0
1
15
1
1
1
1
DES S-box 4 is not like the others
Thanks
Thank you!